FESeries::Legendre< dim, spacedim > Class Template Reference

#include <deal.II/fe/fe_series.h>

Inheritance diagram for FESeries::Legendre< dim, spacedim >:

Public Types
using	CoefficientType = double

Public Member Functions
	Legendre (const std::vector< unsigned int > &n_coefficients_per_direction, const hp::FECollection< dim, spacedim > &fe_collection, const hp::QCollection< dim > &q_collection, const unsigned int component=numbers::invalid_unsigned_int)
	Legendre (const unsigned int n_coefficients_per_direction, const hp::FECollection< dim, spacedim > &fe_collection, const hp::QCollection< dim > &q_collection)
template<typename Number >
void	calculate (const ::Vector< Number > &local_dof_values, const unsigned int cell_active_fe_index, Table< dim, CoefficientType > &legendre_coefficients)
unsigned int	get_n_coefficients_per_direction (const unsigned int index) const
void	precalculate_all_transformation_matrices ()
template<class Archive >
void	save_transformation_matrices (Archive &ar, const unsigned int version)
template<class Archive >
void	load_transformation_matrices (Archive &ar, const unsigned int version)
bool	operator== (const Legendre< dim, spacedim > &legendre) const

Private Attributes
const std::vector< unsigned int >	n_coefficients_per_direction
SmartPointer< const hp::FECollection< dim, spacedim > >	fe_collection
const hp::QCollection< dim >	q_collection
std::vector< FullMatrix< CoefficientType > >	legendre_transform_matrices
std::vector< CoefficientType >	unrolled_coefficients
const unsigned int	component

Subscriptor functionality
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.
std::atomic< unsigned int >	counter
std::map< std::string, unsigned int >	counter_map
std::vector< std::atomic< bool > * >	validity_pointers
const std::type_info *	object_info
void	subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const
void	unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const
unsigned int	n_subscriptions () const
template<typename StreamType >
void	list_subscribers (StreamType &stream) const
void	list_subscribers () const
template<class Archive >
void	serialize (Archive &ar, const unsigned int version)
using	map_value_type = decltype(counter_map)::value_type
using	map_iterator = decltype(counter_map)::iterator
static std::mutex	mutex
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)
static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)
void	check_no_subscribers () const noexcept

Detailed Description

template<int dim, int spacedim = dim>
class FESeries::Legendre< dim, spacedim >

A class to calculate expansion of a scalar FE (or a single component of vector-valued FE) field into series of Legendre functions on a reference element.

Legendre functions are solutions to Legendre's differential equation

\[ \frac{d}{dx}\left([1-x^2] \frac{d}{dx} P_n(x)\right) + n[n+1] P_n(x) = 0 \]

and can be expressed using Rodrigues' formula

\[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}[x^2-1]^n. \]

These polynomials are orthogonal with respect to the \( L^2 \) inner product on the interval \( [-1;1] \)

\[ \int_{-1}^1 P_m(x) P_n(x) = \frac{2}{2n + 1} \delta_{mn} \]

and are complete. A family of \( L^2 \)-orthogonal polynomials on \( [0;1] \) can be constructed via

\[ \widetilde P_m = \sqrt{2} P_m(2x-1). \]

An arbitrary scalar FE field on the reference element \( [0;1] \) can be expanded in the complete orthogonal basis as

\[ u(x) = \sum_{m} c_m \widetilde P_{m}(x). \]

From the orthogonality property of the basis, it follows that

\[ c_m = \frac{2m+1}{2} \int_0^1 u(x) \widetilde P_m(x) dx . \]

This class calculates coefficients \( c_{\bf k} \) using \( dim \)-dimensional Legendre polynomials constructed from \( \widetilde P_m(x) \) using tensor product rule.

Definition at line 272 of file fe_series.h.

Member Typedef Documentation

CoefficientType

template<int dim, int spacedim = dim>

using FESeries::Legendre< dim, spacedim >::CoefficientType = double

Definition at line 275 of file fe_series.h.

Constructor & Destructor Documentation

Legendre() [1/2]

template<int dim, int spacedim>

FESeries::Legendre< dim, spacedim >::Legendre	(	const std::vector< unsigned int > &	n_coefficients_per_direction,
		const hp::FECollection< dim, spacedim > &	fe_collection,
		const hp::QCollection< dim > &	q_collection,
		const unsigned int	component = numbers::invalid_unsigned_int
	)

Constructor that initializes all required data structures.

The n_coefficients_per_direction defines the number of coefficients in each direction, fe_collection is the hp::FECollection for which expansion will be used and q_collection is the hp::QCollection used to integrate the expansion for each FiniteElement in fe_collection.

As the Legendre expansion can only be performed on scalar fields, this class does not operate on vector-valued finite elements and will therefore throw an assertion. However, each component of a finite element field can be treated as a scalar field, respectively, on which Legendre expansions are again possible. For this purpose, the optional parameter component defines which component of each FiniteElement will be used. The default value of component only applies to scalar FEs, in which case it indicates that the sole component is to be decomposed. For vector-valued FEs, a non-default value must be explicitly provided.

Definition at line 199 of file fe_series_legendre.cc.

Legendre() [2/2]

template<int dim, int spacedim>

FESeries::Legendre< dim, spacedim >::Legendre	(	const unsigned int	n_coefficients_per_direction,
		const hp::FECollection< dim, spacedim > &	fe_collection,
		const hp::QCollection< dim > &	q_collection
	)

A non-default constructor. The size_in_each_direction defines the number of coefficients in each direction, fe_collection is the hp::FECollection for which expansion will be used and q_collection is the hp::QCollection used to integrate the expansion for each FiniteElement in fe_collection.

Deprecated:

Use a different constructor instead.

Definition at line 235 of file fe_series_legendre.cc.

Member Function Documentation

calculate()

template<int dim, int spacedim>

template<typename Number >

void FESeries::Legendre< dim, spacedim >::calculate	(	const ::Vector< Number > &	local_dof_values,
		const unsigned int	cell_active_fe_index,
		Table< dim, CoefficientType > &	legendre_coefficients
	)

Calculate legendre_coefficients of the cell vector field given by local_dof_values corresponding to FiniteElement with cell_active_fe_index .

Definition at line 296 of file fe_series_legendre.cc.

get_n_coefficients_per_direction()

template<int dim, int spacedim>

unsigned int FESeries::Legendre< dim, spacedim >::get_n_coefficients_per_direction

(

const unsigned int

index

)

const

Return the number of coefficients in each coordinate direction for the finite element associated with index in the provided hp::FECollection.

Definition at line 285 of file fe_series_legendre.cc.

precalculate_all_transformation_matrices()

template<int dim, int spacedim>

void FESeries::Legendre< dim, spacedim >::precalculate_all_transformation_matrices

Calculate all transformation matrices to transfer the finite element solution to the series expansion representation.

These matrices will be generated on demand by calling calculate() and stored for recurring purposes. Usually, this operation consumes a lot of workload. With this function, all matrices will be calculated in advance. This way, we can separate their costly generation from the actual application.

Definition at line 265 of file fe_series_legendre.cc.

save_transformation_matrices()

template<int dim, int spacedim = dim>

template<class Archive >

void FESeries::Legendre< dim, spacedim >::save_transformation_matrices	(	Archive &	ar,
		const unsigned int	version
	)

Write all transformation matrices of this object to a stream for the purpose of serialization.

Since any of its transformation matrices has to be generated only once for a given scenario, it is common practice to determine them in advance calling precalculate_all_transformation_matrices() and keep them via serialization.

load_transformation_matrices()

template<int dim, int spacedim = dim>

template<class Archive >

void FESeries::Legendre< dim, spacedim >::load_transformation_matrices	(	Archive &	ar,
		const unsigned int	version
	)

Read all transformation matrices from a stream and recover them for this object.

operator==()

template<int dim, int spacedim>

bool FESeries::Legendre< dim, spacedim >::operator== ( const Legendre< dim, spacedim > &  legendre ) const

inline

Test for equality of two series expansion objects.

Definition at line 250 of file fe_series_legendre.cc.

Member Data Documentation

n_coefficients_per_direction

template<int dim, int spacedim = dim>

const std::vector<unsigned int> FESeries::Legendre< dim, spacedim >::n_coefficients_per_direction

private

Number of coefficients in each direction for each finite element in the registered hp::FECollection.

Definition at line 376 of file fe_series.h.

fe_collection

template<int dim, int spacedim = dim>

SmartPointer<const hp::FECollection<dim, spacedim> > FESeries::Legendre< dim, spacedim >::fe_collection

private

hp::FECollection for which transformation matrices will be calculated.

Definition at line 381 of file fe_series.h.

q_collection

template<int dim, int spacedim = dim>

const hp::QCollection<dim> FESeries::Legendre< dim, spacedim >::q_collection

private

hp::QCollection used in calculation of transformation matrices.

Definition at line 386 of file fe_series.h.

legendre_transform_matrices

template<int dim, int spacedim = dim>

std::vector<FullMatrix<CoefficientType> > FESeries::Legendre< dim, spacedim >::legendre_transform_matrices

private

Transformation matrices for each FiniteElement.

Definition at line 391 of file fe_series.h.

unrolled_coefficients

template<int dim, int spacedim = dim>

std::vector<CoefficientType> FESeries::Legendre< dim, spacedim >::unrolled_coefficients

private

Auxiliary vector to store unrolled coefficients.

Definition at line 396 of file fe_series.h.

component

template<int dim, int spacedim = dim>

const unsigned int FESeries::Legendre< dim, spacedim >::component

private

Which component of FiniteElement should be used to calculate the expansion.

Definition at line 402 of file fe_series.h.

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The documentation for this class was generated from the following files:

• include/deal.II/fe/fe_series.h
• source/fe/fe_series_legendre.cc